by Pasquale Pavone & Konstantin Lion for exciting fluorine
Purpose: In this tutorial you will learn how to set up and execute a series of calculations for strained structures. Additionally, it will be explained how to obtain the derivatives of the energy-vs-strain curves at zero strain and how these quantities are related to elastic constants.
Table of Contents
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0. Define relevant environment variables
Read the following paragraphs before starting with the rest of this tutorial!
Before starting, be sure that relevant environment variables are already defined as specified in How to set environment variables for tutorials scripts. Here is a list of the scripts which are relevant for this tutorial with a short description.
- SETUP-elastic-strain.py: Python script for generating strained structures.
- EXECUTE-elastic-strain.sh: (Bash) shell script for running a series of exciting calculations.
- CHECKFIT-energy-vs-strain.py: Python script for extracting derivatives at zero strain of energy-vs-strain curves.
- PLOT-energy.py: Python visualization tool for energy-vs-strain curves.
- PLOT-status.py: Python visualization tool for RMS deviations of the SCF potential as a function of the iteration number during the SCF loop.
- PLOT-maxforce.py: Python visualization tool for the maximum amplitude of the force on the atoms during relaxation.
- PLOT-checkderiv.py: Python visualization tool for the calculation of derivatives at zero strain using the fit of energy-vs-strain curves.
- PLOT-optimized-geometry.py: Python visualization tool for relaxed coordinates of atoms in the unit cell.
From now on the symbol $ will indicate the shell prompt.
Requirements: Bash shell. Python numpy, lxml, matplotlib.pyplot, and sys libraries.
1. Theoretical background
2. Set up the calculations
i) Preparation of the input file
The first step is to create a directory for each system that you want to investigate. Here, we consider the calculation of the energy-vs-strain curves for carbon in the diamond structure. However, the procedure we show you is valid for any system. Thus, we will create a directory diamond-elastic-strain and we move inside it.
$ mkdir diamond-elastic-strain
$ cd diamond-elastic-strain
Inside this directory, we create (or copy from a previous calculation) the file input.xml corresponding to a calculation for the equilibrium structure of diamond. This file could look like the following.
<input> <title>Diamond: Equilibrium structure</title> <structure speciespath="$EXCITINGROOT/species"> <crystal scale="6.714"> <basevect> 0.5 0.5 0.0 </basevect> <basevect> 0.5 0.0 0.5 </basevect> <basevect> 0.0 0.5 0.5 </basevect> </crystal> <species speciesfile="C.xml" rmt="1.25"> <atom coord="0.00 0.00 0.00" /> <atom coord="0.25 0.25 0.25" /> </species> </structure> <groundstate ngridk="8 8 8" swidth="0.0001" gmaxvr="14" xctype="GGA_PBE_SOL"> </groundstate> <relax/> </input>
Please, remember that the input file for an exciting calculation must always be called input.xml.
Be sure to set the correct path for the exciting root directory (indicated in this example by $EXCITINGROOT) to the one pointing to the place where the exciting directory is placed. In order to do this, use the command
$ SETUP-excitingroot.sh
Be sure to have in your file the appropriate command for performing the structure optimization: Deforming your system may change the relative positions of the atoms in the unit cell.
<relax/>
ii) Generation of input files for distorted structures
All strains considered in this tutorial are Lagrangian strains.
In order to generate input files for a series of distorted structure, you have to run the script SETUP-elastic-strain.py. Notice that the script SETUP-elastic-strain.py always generates a working directory containing input files for different strains. Results of the current calculations will be also stored in the working directory. The directory name can be specified by adding the name in the command line.
$ SETUP-elastic-strain.py DIRECTORYNAME
If no name is given, the script use the default name workdir. Very important: The working directory is overwritten each time you execute the script SETUP-elastic-strain.py. Therefore, choose different names for different calculations.
The script SETUP-elastic-strain.py produces the following output on the screen (using deformation-0 as working directory).
$ SETUP-elastic-strain.py deformation-0
Enter maximum Lagrangian strain [smax] >>>> 0.10
Enter the number of strain values in [-smax,smax] >>>> 11
------------------------------------------------------------------------
List of deformation codes for strains in Voigt notation
------------------------------------------------------------------------
0 => ( eta, eta, eta, 0, 0, 0) | volume strain
1 => ( eta, 0, 0, 0, 0, 0) | linear strain along x
2 => ( 0, eta, 0, 0, 0, 0) | linear strain along y
3 => ( 0, 0, eta, 0, 0, 0) | linear strain along z
4 => ( 0, 0, 0, eta, 0, 0) | yz shear strain
5 => ( 0, 0, 0, 0, eta, 0) | xz shear strain
6 => ( 0, 0, 0, 0, 0, eta) | xy shear strain
7 => ( 0, 0, 0, eta, eta, eta) | shear strain along (111)
8 => ( eta, eta, 0, 0, 0, 0) | xy in-plane strain
9 => ( eta, -eta, 0, 0, 0, 0) | x(-y) in-plane strain
10 => ( eta, eta, eta, eta, eta, eta) | global strain
11 => ( eta, 0, 0, eta, 0, 0) | mixed strain
12 => ( eta, 0, 0, 0, eta, 0) | mixed strain
13 => ( eta, 0, 0, 0, 0, eta) | mixed strain
14 => ( eta, eta, 0, eta, 0, 0) | mixed strain
------------------------------------------------------------------------
Enter deformation code >>>> 0
$
In this example, (on screen) input entries are preceded by the symbol ">>>>". Entry values must be typed on the screen when requested. The first entry (in our example 0.10) represents the absolute value of the maximum strain for which we want to perform the calculation. The second entry (11) is the number of deformed structures equally spaced in strain, which are generated between the maximum negative strain and the maximum positive one. The third (last) entry (0) is a self-explained label indicating the type of deformation. The latter is always referred to 2-dimensional strain tensors in the Voigt notation (so that, e.g., a strain value of 0.10 corresponds, for the choice 1 of the deformation code, to a linear deformation of 10% along the x direction).
After running the script, a directory called deformation-0 is created, which contains input files for different strain values.
3. Execute the calculations
To execute the series of calculation with input files created by SETUP-elastic-strain.py you have to run the script EXECUTE-elastic-strain.sh. If a name for the working directory has been specified, then you must give it here, too.
$ EXECUTE-elastic-strain.sh deformation-0
===> Output directory is "deformation-0" <===
Running exciting for file input-01.xml ----------------------------------
...
Run completed for file input-11.xml -------------------------------------
$
After the complete run, move to the working directory deformation-0.
$ cd deformation-0
Inside this directory, results of the calculation for the input file input-i.xml are contained in the subdirectory rundir-i where i is running from 01 to the total number of strain values. The data for energy-vs-strain curves are contained in the file energy-vs-strain.
4. Post-processing: Extract energy derivatives
At this point, inside the directory deformation-0, you can use the python script CHECKFIT-energy-vs-strain.py for extracting derivatives at zero strain of energy-vs-strain curves.
$ CHECKFIT-energy-vs-strain.py
Enter maximum strain for the fit >>>> 0.10
Enter the order of derivative >>>> 2
###########################################
Fit data-----------------------------------
Deformation code ==> 0
Deformation label ==> EEE000
Maximum value of the strain ==> 0.10000000
Number of strain values used ==> 11
Fit results for the derivative of order 2
Polynomial of order 2 ==> 4467.23 [GPa]
Polynomial of order 3 ==> 4467.23 [GPa]
Polynomial of order 4 ==> 4053.47 [GPa]
Polynomial of order 5 ==> 4053.47 [GPa]
Polynomial of order 6 ==> 4060.24 [GPa]
Polynomial of order 7 ==> 4060.24 [GPa]
###########################################
$
In this example, input entries are preceded by the symbol ">>>>". Entry values must be typed on the screen when requested. The first entry (in our example 0.10) represents the absolute value of the maximum strain for which we want to perform the calculation. The second entry (2) is the order of the derivative that we want to obtain.
The script generates the output files check-energy-derivatives and order-of-derivative, which can be used in the post-processing analysis. Results of this script can be analyzed using the visualization tool PLOT-checkderiv.py.
5. Post-processing: Visualization tools
All the scripts mentioned here must be executed in the directory where the energy-vs-strain, check-energy-derivatives, and order-of-derivative files are located. The scripts produce as output a PostScript file named PLOT.ps as well as a png file (PLOT.png).
i) PLOT-energy.py
This script allows for the visualization of the energy-vs-strain curve. It is executed as follows.
$ PLOT-energy.py
ii) PLOT-checkderiv.py
This is a very important tool that allows to represent the dependence of the calculated derivatives of the energy-vs-strain curve on
- the range of points included in the fitting procedure ("maximum lagrangian strain"),
- the maximum degree of the polynomial used in the fitting procedure ("n").
The script PLOT-checkderiv.py requires as input the check-energy-derivatives and order-of-derivative files generated by CHECKFIT-energy-vs-strain.py and is executed as follows.
$ PLOT-checkderiv.py YMIN YMAX
The previous plots can be used to determine the best range of deformations and order of polynomial fit for each distortion. By analyzing the plot, we note that curves corresponding to the higher order of the polynomial used in the fit show a horizontal plateau at about 4060 GPa. This can be assumed to be the converged value for the second derivative, from the point of view of the fit (further information on this topic can be found here). For this distortion type, this value equals 9 times the bulk modulus. Thus, the extracted value of the bulk modulus is about 451 GPa.
iii) PLOT-status.py
Python visualization tool for the RMS deviations of the effective SCF potential as a function of the iteration number during the SCF loop. It is executed as follows.
$ PLOT-status.py LABEL
Different line segments correspond to SCF calculations for different geometries during the relaxation.
iv) PLOT-maxforce.py
Python visualization tool for the maximum amplitude of the force on atoms during relaxation. It is useful for deformations which allow for internal relaxation of atomic positions, e.g., for the deformation with the code 7. It is executed as follows.
$ PLOT-maxforce.py LABEL
The input entry definition is the same as for the script PLOT-status.py. If the symmetry of the deformation applied to the crystal is such that no extra force is applied to the atoms (e.g., as it happens for deformation 1 and 2) the output of the script PLOT-maxforce.py will be
Either data file not (yet) ready for visualization
or maximum force target reached already at the initial configuration.
The red points show the calculated value at each optimization step, whereas the blue line indicates the target value of the maximum amplitude of the force for stopping the relaxation.
v) PLOT-optimized-geometry.py
Python visualization tool for showing the optimized geometry compared to the reference (unrelaxed) geometry for the relative atomic coordinates of two atoms in the unit cell as a function of Lagrangian strain. It is useful for deformations which allow for internal relaxation of atomic positions, e.g., for the deformation with the code 7. It is executed as follows.
$ PLOT-optimized-geometry.py ATOM1 ATOM2 YMIN YMAX
Here, (Δ1, Δ2, Δ3) and (Δ1ref, Δ2ref, Δ3ref) represent the position difference vector, rATOM2 - rATOM1, expressed in lattice coordinates, for the optimized geometry and the unrelaxed (reference) case, respectively.
6. Post-processing: How to derive elastic constants
Second derivatives calculated at zero strain of energy-vs-strain curves are combinations of the elastic constants Cij where the indexes i,j=1,2,…,6 are given in the Voigt notation. In the example that we are considering here, carbon in the cubic diamond structure, only 3 different elastic constants are non vanishing
- C11
- C12
- C44
In order to extract these three elastic constants, three different deformation types must be used. For cubic systems the best choice is represented by the following deformation types
- Volume strain (in our script corresponding to the label 0)
- Uniaxial strain in the 100 direction (label 1)
- Shear strain along the 111 direction (label 7)
Which in turns correspond to the following combination of elastic constants:
- label 0: 3 C11+ 6 C12 = 9 B0
- label 1: C11
- label 7: 3 C44
where B0 is the bulk modulus.
Experimental reference values for diamond:
- C11 = 1076 GPa
- C12 = 125 GPa
- C44 = 577 GPa
- B0 = 452 GPa